Kim, Djun Maximilian; Rolfsen, Dale An ordering for groups of pure braids and fibre-type hyperplane arrangements. (English) Zbl 1047.20027 Can. J. Math. 55, No. 4, 822-838 (2003). The braid group \(B_n\) admits a presentation with generators \(\sigma_1,\dots,\sigma_{n-1}\) and relations \(\sigma_i\sigma_j=\sigma_j\sigma_i\) if \(|i-j|>1\) for all \(i,j=1,\dots,n-1\) and \(\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}\) for all \(i=1,\dots,n-2\). The pure braids form a normal subgroup \(P_n\) of index \(n!\). P. Dehornoy [J. Knot Theory Ramifications 4, No. 1, 33-79 (1995; Zbl 0873.20030] proved that the group \(B_n\) admits one sided orders but it is not two sided orderable (bi-orderable in the terminology used by the authors) if \(n\geq 3\). On the other hand, \(P_n\) admits two sided orders and the authors exhibit one such total order on \(P_n\) (and also of the fibre-type hyperplane arrangement groups) using the fact that such groups are residually torsion-free nilpotent. The paper contains structural description of \(P_n\) from several points of view. Reviewer: Akbar H. Rhemtulla (Edmonton) Cited in 14 Documents MSC: 20F36 Braid groups; Artin groups 20F60 Ordered groups (group-theoretic aspects) 06F15 Ordered groups 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Keywords:braid groups; ordered groups; free groups; right orders; pure braids; hyperplane arrangements Citations:Zbl 0873.20030 PDFBibTeX XMLCite \textit{D. M. Kim} and \textit{D. Rolfsen}, Can. J. Math. 55, No. 4, 822--838 (2003; Zbl 1047.20027) Full Text: DOI